Inter-modal interference of arbitrary orbital angular momentum modes in nonlinear frequency conversion

Abstract

Nonlinear interaction between orbital angular momentum (OAM) modes provides an enticing promise for realizing high-dimensional quantum states and multimodal communications. Here, we demonstrate the inter-modal interference of OAM modes based on the nonlinear frequency conversion of perfect vortex beams, permitting the arbitrary spatial intensity and phase information to be transformed and manipulated from 778 nm input beam to 420 nm output beam. The conversion of single OAM state obeys the typical arithmetic operation. In contrast, the conversion of superposition state is predicted and revealed as the inter-modal interference of full spatial complex amplitude. For the allowed OAM modes, the probability amplitude is quantitatively analyzed by the intensity overlap of input OAM modes, which determines the degree of degradation in the intensity and phase. As a result, the output beam exhibits the nonuniform petal-shaped intensity and phase distributions under the conditions of phase matching and OAM conservation. Furthermore, the inherent transverse structure invariance of perfect vortex beams enables them to accommodate and support larger OAM values and more superposition states.

Graphical abstract

Keywords

orbital angular momentum / nonlinear frequency conversion / inter-modal interference

1 Introduction

The quantized orbital angular momentum (OAM), defined in an infinite-dimensional Hilbert space [1], has attracted significant attention in quantum information processing [2, 3], super-resolution imaging [4, 5] and optical tweezer technology [6, 7]. Among these, nonlinear frequency conversion of OAM has emerged as a crucial photonic technique for utilizing the spatial and frequency domains simultaneously [8−11], which is rapidly leading to advances in mode detection [12, 13], frequency interface [14, 15] and nonlinear holography [16−18]. Corresponding studies in the area begin with the conservation of OAM [19−22], which is mainly focused on the direct information transport and simple arithmetic operation of OAM states [23−25], including the addition, subtraction, and cancellation.

Recently, the superposition states composed of several OAM components provide a possible for further enhancing the information capacity. In the context of high-dimensional quantum information [26, 27], the high-fidelity frequency conversion of multi-dimensional superposition states has been demonstrated [28−30] and subsequently implemented for remote transport [31]. Notably, compared to the single OAM state, the nonlinear interaction of superposition states yields the novel spatial modes and encodes more abundant information owing to the inter-modal interference effect [32]. However, it remains a challenge in the interaction of OAM components stemming from the complex selection rule [33] and intrinsic amplitude dependence [34].

The perfect vortex beam (PVB), as the Fourier transformation of Bessel beam, exhibits the topological-charge-independent transverse structure [35]. This unique property enables the ideal mode matching for superposition states composed of OAM components with arbitrary quantity and value, supplying the foundation for the nonlinear interaction of OAM states [36]. It is because that the coherent superposition of multiple OAM states is directly associated with the transverse structure of the OAM components, resulting in the mutual spatial-independence for traditional vortex beams with different topological charges. The introduction of PVB with transverse structure invariance effectively circumvents the limitation and opens up a new avenue for the transfer and manipulation of OAM [37, 38].

In this work, we propose and demonstrate the inter-modal interference based on the nonlinear frequency conversion of PVB, which is implemented through the monochromatic two-photon transition of ⁸⁵Rb atoms. The 778 nm PVB, combining with the internal 5233 nm infrared beam, induces the four-wave mixing (FWM) process to generate the 420 nm output beam. The spatial intensity and phase structure of the output beam are determined by probability amplitudes of all permissible OAM components, which is quantitatively analyzed through the intensity overlap. For the single OAM state of 778 nm input beam, the typical arithmetic operation is observed in the conversion process, which obeys the selection rule l₄₂₀ = 2l₇₇₈. Once superposition states are used, the 420 nm output beam exhibits the nonuniform petal-shaped intensity and phase distributions due to the inter-modal interference effect.

2 Experimental setup

Figure 1(a) shows the diamond-type energy level system of ⁸⁵Rb atoms, which is consisted of 5S_1/2, 5P_3/2, 5D_5/2, and 6P_3/2 states. The monochromatic two-photon transition is driven by the simultaneous absorptions of two 778 nm photons, exciting ⁸⁵Rb atoms from the 5S_1/2 ground state to the 5D_5/2 excited state via a virtual state. Then, the atoms decay from the 5D_5/2 state to the 6P_3/2 state, leading to the generation of the third optical field at 5233 nm and inducing the FWM process. Finally, the 420 nm output beam is efficiently generated when the optimal phase-matching condition is satisfied, resulting in the transfer and manipulation of OAM. This mechanism not only simplifies the experimental setup, but also eliminates extraneous phase differences in contrast to the traditional two lasers excitation systems. The schematic of the inter-modal interference in the conversion process is illustrated in Fig. 1(b), which reveals the selection rule of spatial modes. Under the restriction of OAM conservation, the topological charges of four light fields are satisfied by l₇₇₈ + l₇₇₈ = l₄₂₀ + l₅₂₃₃, here l₅₂₃₃ = 0 is assumed. Thus, for the input beam of \left({\mathrm{\Phi }}_{778}^l+{\mathrm{\Phi }}_{778}^{-l}\right)/\sqrt{2}, the output beam comprises three possible modes {\mathrm{\Phi }}_{420}^{2l}, {\mathrm{\Phi }}_{420}^{-2l}, and {\mathrm{\Phi }}_{420}^0 with the different probability amplitude u_i, corresponding spatial intensity and phase distributions depend on the interference and superposition of all OAM components.

The experimental setup is illustrated in Fig. 1(c), which is consistent with previous works [39−41]. The 778 nm pump laser is provided by a tapered amplifier diode laser (DLC TA pro, Toptica), and the single-mode fiber is used to filter the spatial mode and ensures the output of fundamental Gaussian beam (GB). The GB with the beam waist w_g of about 0.5 mm is transformed into a Bessel−Gaussian beam (BGB) through a vortex retarder (VR) and an axicon (AX), where the refractive index and base angle of AX are n=1.5 and \gamma ={0.5}^{\circ } respectively. Then, a lens (L1) with the focal length of f_1=50\mathrm{m}\mathrm{m} is employed to perform the Fourier transform of BGB, and generates the PVB at the rear focal plane. The prepared 778 nm PVB is imaged into the center of a 2 cm long vapor cell by a 4-f system consisting of L2 (f_2=100\mathrm{m}\mathrm{m}) and L3 (f_3=50\mathrm{m}\mathrm{m}), and the corresponding ring radius and ring half-width are calculated as r_0=0.5f_1(n-1)\sin \gamma and w_0=f_1/(k{w}_g). The quarter-wave plate and polarization beam splitter (PBS) are utilized to switch the single OAM states to superposition states. When the phase matching is satisfied, the 420 nm output beam is effectively generated via FWM process. Here, the output power of about 14 μW is obtained under the input power of 600 mW. The temperature of vapor cell is precisely kept at 190 °C by a self-feedback system, corresponding to the atomic density of 6.14\times 10^{14}\mathrm{c}{\mathrm{m}}^{-3}. Finally, the 420 nm output beam is separated from the background light using an interference filter (center wavelength 420 nm, 10 nm passband), and recorded by a charge-coupled device (CCD).

3 Results and discussion

The FWM process based on the monochromatic two-photon transition can be described by the overlap of the involved light fields E_{778}^{\ast }{E}_{778}^{\ast }{E}_{420}{E}_{5233}, which determines the probability amplitude of each output mode. Considering the Fresnel diffraction formula, the free space propagation of PVB is represented as [42]

(1)\begin{array}{rl}{\mathrm{\Phi }}^l\left(r,\theta ,z\right)=& A_0\frac{w_0}{w\left(z\right)}(-1{)}^l\exp \left(\mathrm{i}\psi +\mathrm{i}l\theta +\mathrm{i}kz\right)\\ & \cdot \exp \left\{-\frac{1}{w(z{)}^2}\left[r^2-{\left(\frac{r_{0}z}{z_r}\right)}^2\right]\right\}\\ & \cdot I_l\left[\frac{2r{r}_0\exp (\mathrm{i}\psi )}{w\left(z\right)w_0}\right],\end{array}

where A₀ is the constant amplitude, r_0 and w_0 are the ring radius and ring half-width of PVB, respectively, l and I_l are the topological charge and modified l-th order Bessel function of first kind, respectively, k=2\pi /\lambda, w(z)=w_0\sqrt{[1+(z/z_r{)}^2]}, \psi =\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(z/z_r) and z_r=\pi w_0^2/\lambda are the wave vector, beam radius at different position z, Gouy phase and Rayleigh range. Then, the probability amplitude under each specific combination of spatial mode components \mathrm{\Phi } is given by [32, 43]

(2)u={\displaystyle {\int }_{-L/2}^{L/2}{\displaystyle {\int }_0^R{\displaystyle {\int }_0^{2\pi }r{\mathrm{\Phi }}_{778}^{\ast }{\mathrm{\Phi }}_{778}^{\ast }{\mathrm{\Phi }}_{420}{\mathrm{\Phi }}_{5233}\mathrm{d}\theta \mathrm{d}r\mathrm{d}z,}}}

where L is the length of the vapor cell and R=\lambda /(\pi w_0) is the numerical aperture of the system. The corresponding probability of respective modes is u^2. Notably, the output modes should exhibit as PVBs, and l₅₂₃₃ = 0 is mainly expected and assumed because that the 5233 nm light field is internally generated via amplified spontaneous emission [44−46].

Figure 2 shows the theoretically simulated intensity profile and phase of input and output beams for different OAM states. For the OAM states E_{778}={\mathrm{\Phi }}_{778}^l, the 420 nm output beams only exhibit a single OAM mode of {\mathrm{\Phi }}_{420}^{2l}, which is the typical arithmetic operation of OAM, as shown in the first and second lines. On the contrary, under the case of superposition states E_{778}=({\mathrm{\Phi }}_{778}^l+{\mathrm{\Phi }}_{778}^{-l})/\sqrt{2}, the 420 nm output beam contains three possible OAM modes of {\mathrm{\Phi }}_{420}^{2l}, {\mathrm{\Phi }}_{420}^{-2l}, and {\mathrm{\Phi }}_{420}^0, expressed as

(3)E_{420}=u_1{\mathrm{\Phi }}_{420}^{2l}+u_2{\mathrm{\Phi }}_{420}^{-2l}+u_3{\mathrm{\Phi }}_{420}^0,

where the normalized probability amplitudes u_i depend on the mode overlap from Eq. (2), and the mode evolution caused by the Gouy phase is ignored, owing to its intrinsic independence from the topological charge compared to the traditional vortex beam. Based on the interference principle of OAM modes, the 420 nm output beams exhibit the petal-shaped intensity and phase distributions, as shown in the third and fourth lines. Thus, the inter-modal interference of all components would lead to more abundant intensity and phase information.

In order to quantitatively analyze the nonlinear interaction between OAM modes, the intensity and phase of all possible OAM modes in 420 nm output beam are theoretically simulated in Fig. 3. The first line to the fourth line are the cases of \left({\mathrm{\Phi }}_{778}^1+{\mathrm{\Phi }}_{778}^{-1}\right)/\sqrt{2}, ({\mathrm{\Phi }}_{778}^2+{\mathrm{\Phi }}_{778}^{-2})/\sqrt{2}, \left({\mathrm{\Phi }}_{778}^3+{\mathrm{\Phi }}_{778}^{-3}\right)/\sqrt{2} and \left({\mathrm{\Phi }}_{778}^4+{\mathrm{\Phi }}_{778}^{-4}\right)/\sqrt{2}, corresponding output modes E_{420} are calculated as

E_{420}=\begin{array}{c}0.46{\mathrm{\Phi }}_{420}^2+0.46{\mathrm{\Phi }}_{420}^{-2}+0.76{\mathrm{\Phi }}_{420}^0,\\ 0.43{\mathrm{\Phi }}_{420}^4+0.43{\mathrm{\Phi }}_{420}^{-4}+0.79{\mathrm{\Phi }}_{420}^0,\\ 0.66{\mathrm{\Phi }}_{420}^6+0.66{\mathrm{\Phi }}_{420}^{-6}-0.37{\mathrm{\Phi }}_{420}^0,\\ 0.69{\mathrm{\Phi }}_{420}^8+0.69{\mathrm{\Phi }}_{420}^{-8}+0.24{\mathrm{\Phi }}_{420}^0.\end{array}

Obviously, u_1=u_2 always holds true due to the symmetry of OAM modes, and the output mode should exhibit the 4l-petaled bright spots in the absence of the third mode component {\Phi }_{420}^0. However, the existence of {\Phi }_{420}^0 leads to the inter-modal interference, and the probability amplitude u_3 ultimately determines the intensity and phase of 420 nm output beam, as illustrated in Fig. 3(e). The interference effect can be obtained from the typical interference of plane waves E_i=u_i{\mathrm{e}}^{\mathrm{i}{\phi }_i}, where u_i and {\phi }_i denote the amplitude and phase of the light field, respectively. For the two light fields E_1 and E_2, the constructive and destructive interference conditions are given by {\mathrm{\Delta }\phi }_{12}=2n\pi and {\mathrm{\Delta }\phi }_{12}=(2n+1)\pi, n\in \mathbb{Z}, where {\mathrm{\Delta }\phi }_{ij} represents the relative phase and u_1=u_2 is assumed. Furthermore, the interference intensity of three light fields is given by I={\left|u_1+u_2\right|}^2+{\left|u_3\right|}^2+2(u_1+u_2)u_3\cos \mathrm{\Delta }\varphi, where \cos \mathrm{\Delta }\varphi =\cos {\mathrm{\Delta }\phi }_{13}=\cos {\mathrm{\Delta }\phi }_{23} is satisfied under the case of {\mathrm{\Delta }\phi }_{12}=2n\pi. The corresponding constructive and destructive interference conditions are u_3\cos \mathrm{\Delta }\varphi =\left(u_1+u_2\right) and u_3\cos \mathrm{\Delta }\varphi =-\left(u_1+u_2\right) when {|u}_3|=\left|u_1+u_2\right|. The similar derivations can be extended to the spatial modes carrying with OAM. The u_3 with a positive (negative) value represents the phase delay of zero (\pi ), which leads to the destructive interference and constructive interference for the petaled bright spots with the phase of \pi (zero) and zero (\pi). Hence, the 4l-petaled bright spots degrade to the 2l-petaled bright spots since the {|u}_3| is close to \left|u_1+u_2\right| for the former two modes. In contrast, for the latter two modes, the difference between the {|u}_3| and \left|u_1+u_2\right| is significant, the spatial intensity of 420 nm output beam exhibits nonuniform 4l-petaled bright spots as a result of partial destructive interference.

Figure 4 shows the observed intensity profile of input and output beams for the experimental verification. Here, the single OAM state and superposition state are generated and converted by switching the polarization states of initial GB, corresponding to the circularly and linearly polarization. Figure 4(a) illustrates the arithmetic operations of single OAM states, which demonstrates that the 420 nm output beam carries twice the OAM of the 778 nm input beam while maintains a stable conversion efficiency independent of OAM. The topological charge is detected using a tilted lens and displayed in the inset, which is based on astigmatic transformation of vortex beam. Figure 4(b) shows the frequency conversion of superposition states, which excellently agrees with theoretical simulation. The direct measurement of weight u^2 is currently constrained by the absence of spatial light modulator (SLM) at 420 nm wavelength band. Thus, the validation is implemented through interference results, which are determined by the weights of OAM components. The ratio of the average intensity is defined as \beta =\overline{I_{max}^D}/\overline{I_{max}^C} to characterize the interference effect of multiple OAM components, where I_{max}^D and I_{max}^C are the maximum intensity of each destructive interference and constructive interference spot in output beams. For the former two modes, the theoretical and experimental ratios of average intensity are calculated as {\beta }_1^{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{y}}={\beta }_2^{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{y}}={\beta }_1^{\exp }\approx {\beta }_2^{\exp }\approx0 due to the complete destructive interference effect. For the latter two modes, the theoretical average intensity ratios are calculated as {\beta }_3^{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{y}}=0.32 and {\beta }_4^{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{y}}=0.49, the corresponding experimental results are approximately {\beta }_3^{\exp }\approx 0.34 and {\beta }_4^{\exp }\approx 0.48. The inter-modal interference plays a critical role in the transfer and manipulation of OAM, which can be quantitatively predicted by evaluating the overlap integral of Eq. (2).

Although only the superposition state of ({\mathrm{\Phi }}_{778}^l+{\mathrm{\Phi }}_{778}^{-l})/\sqrt{2} is used in our experiment, this method also can be generalized to arbitrary OAM states due to the transverse structure invariance of PVB. Figure 5 simulates the nonlinear frequency conversion of superposition states composed of three OAM components. The first line to fourth line are the cases of ({\mathrm{\Phi }}_{778}^1+{\mathrm{\Phi }}_{778}^1+{\mathrm{\Phi }}_{778}^{-3})/\sqrt{3}, \left({\mathrm{\Phi }}_{778}^1+{\mathrm{\Phi }}_{778}^{-2}+{\mathrm{\Phi }}_{778}^3\right)/\sqrt{3}, ({\mathrm{\Phi }}_{778}^1+{\mathrm{\Phi }}_{778}^1+{\mathrm{\Phi }}_{778}^{-4})/\sqrt{3}, and \left({\mathrm{\Phi }}_{778}^1+{\mathrm{\Phi }}_{778}^2+{\mathrm{\Phi }}_{778}^{-4}\right)/\sqrt{3}, corresponding output modes E_{420} are calculated as

\begin{array}{rl}& E_{420}=\\ & \begin{array}{c}0.52{\mathrm{\Phi }}_{420}^2+0.52{\mathrm{\Phi }}_{420}^2+0.20{\mathrm{\Phi }}_{420}^{-6}+0.52{\mathrm{\Phi }}_{420}^2+0.26{\mathrm{\Phi }}_{420}^{-2}+0.26{\mathrm{\Phi }}_{420}^{-2},\\ 0.18{\mathrm{\Phi }}_{420}^2+0.43{\mathrm{\Phi }}_{420}^{-4}+0.25{\mathrm{\Phi }}_{420}^6+0.59{\mathrm{\Phi }}_{420}^{-1}+0.38{\mathrm{\Phi }}_{420}^4+0.47{\mathrm{\Phi }}_{420}^1,\\ 0.47{\mathrm{\Phi }}_{420}^2+0.47{\mathrm{\Phi }}_{420}^2+0.17{\mathrm{\Phi }}_{420}^{-8}+0.47{\mathrm{\Phi }}_{420}^2+0.39{\mathrm{\Phi }}_{420}^{-3}+0.39{\mathrm{\Phi }}_{420}^{-3},\\ 0.18{\mathrm{\Phi }}_{420}^2+0.43{\mathrm{\Phi }}_{420}^4+0.25{\mathrm{\Phi }}_{420}^{-8}+0.65{\mathrm{\Phi }}_{420}^3+0.30{\mathrm{\Phi }}_{420}^{-3}+0.46{\mathrm{\Phi }}_{420}^{-2}.\end{array}\end{array}

It can be observed that the output beam consists of six OAM components and possesses non-zero OAM due to the asymmetry of input OAM components. However, some specific OAM components have extremely small probability amplitudes, which can be approximately neglected in the inter-modal interference effect. This nonlinear frequency conversion of PVB provides an effective approach to generate and manipulate arbitrary OAM states with the assistance of a SLM.

4 Conclusion

In summary, we propose and demonstrate the inter-modal interference of arbitrary OAM modes based on the nonlinear frequency conversion of PVB, which realizes the transfer and manipulation of spatial intensity and phase information. Under the phase matching and OAM conservation conditions, all permissible OAM components in 420 nm output beam are quantitatively analyzed via probability amplitude. The input beam with the single OAM state obeys the arithmetic operation in the conversion process, and leads to the output beam carrying with double OAM. Furthermore, the frequency conversion of superposition state is predicted and revealed via inter-modal interference, which allows the transfer and manipulation of arbitrary spatial intensity and phase information in view of the flexible tunability and transverse structure invariance of PVB. This mechanism of inter-modal interference in the nonlinear process advances the understanding of light-atom interactions and opens a new avenue for the transfer and manipulation of optical modes, especially for the field of high-capacity optical communication networks.

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